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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Modular invariance for conformal full field algebras


Authors: Yi-Zhi Huang and Liang Kong
Journal: Trans. Amer. Math. Soc. 362 (2010), 3027-3067
MSC (2000): Primary 17B69; Secondary 81T40
Published electronically: December 22, 2009
MathSciNet review: 2592945
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Abstract: Let $ V^{L}$ and $ V^{R}$ be simple vertex operator algebras satisfying certain natural uniqueness-of-vacuum, complete reducibility and cofiniteness conditions and let $ F$ be a conformal full field algebra over $ V^{L}\otimes V^{R}$. We prove that the $ q_{\tau}$- $ \overline{q_{\tau}}$-traces (natural traces involving $ q_{\tau}=e^{2\pi i\tau}$ and $ \overline{q_{\tau}}= \overline{e^{2\pi i\tau}}$) of geometrically modified genus-zero correlation functions for $ F$ are convergent in suitable regions and can be extended to doubly periodic functions with periods $ 1$ and $ \tau$. We obtain necessary and sufficient conditions for these functions to be modular invariant. In the case that $ V^{L}=V^{R}$ and $ F$ is one of those constructed by the authors in an earlier paper, we prove that all these functions are modular invariant.


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Additional Information

Yi-Zhi Huang
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854-8019
Email: yzhuang@math.rutgers.edu

Liang Kong
Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103, Leipzig, Germany – and – Institut Des Hautes Études Scientifiques, Le Bois-Marie, 35, Route De Chartres, F-91440 Bures-sur-Yvette, France
Address at time of publication: Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China
Email: kong.fan.liang@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04933-2
PII: S 0002-9947(09)04933-2
Received by editor(s): March 10, 2008
Published electronically: December 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.